Definition:Neighborhood (Real Analysis)

Definition

Let $\alpha \in \R$ be a real number.

On the real number line with the usual metric, the $\epsilon$-neighborhood of $\alpha$ is defined as the open interval:

$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon .. \alpha + \epsilon}\right)$

From the definition of the real numbers as a metric space, it can be seen that this definition is compatible with the definition of a neighborhood in a metric space.

Linguistic Note

The UK English spelling of this is neighbourhood.